Properties

Label 30.2.e
Level 30
Weight 2
Character orbit e
Rep. character \(\chi_{30}(17,\cdot)\)
Character field \(\Q(\zeta_{4})\)
Dimension 4
Newform subspaces 1
Sturm bound 12
Trace bound 0

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Defining parameters

Level: \( N \) = \( 30 = 2 \cdot 3 \cdot 5 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 30.e (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 15 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 1 \)
Sturm bound: \(12\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(30, [\chi])\).

Total New Old
Modular forms 20 4 16
Cusp forms 4 4 0
Eisenstein series 16 0 16

Trace form

\( 4q - 4q^{3} - 4q^{6} - 4q^{7} + O(q^{10}) \) \( 4q - 4q^{3} - 4q^{6} - 4q^{7} + 4q^{10} + 4q^{12} + 8q^{15} - 4q^{16} + 8q^{18} + 8q^{21} - 4q^{22} - 16q^{25} - 4q^{27} - 4q^{28} - 12q^{30} - 8q^{31} - 4q^{33} - 4q^{36} + 24q^{37} + 8q^{40} + 4q^{42} + 24q^{43} - 8q^{45} + 16q^{46} + 4q^{48} - 8q^{51} + 12q^{55} - 16q^{57} - 20q^{58} - 4q^{60} - 24q^{61} - 4q^{63} + 8q^{66} - 16q^{67} + 4q^{70} - 8q^{72} - 20q^{73} + 28q^{75} + 16q^{76} + 28q^{81} - 16q^{82} + 8q^{85} + 20q^{87} - 4q^{88} + 8q^{90} + 8q^{93} + 4q^{96} + 12q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(30, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
30.2.e.a \(4\) \(0.240\) \(\Q(\zeta_{8})\) None \(0\) \(-4\) \(0\) \(-4\) \(q+\zeta_{8}q^{2}+(-1-\zeta_{8}^{2}+\zeta_{8}^{3})q^{3}+\zeta_{8}^{2}q^{4}+\cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{4} \)
$3$ \( 1 + 4 T + 8 T^{2} + 12 T^{3} + 9 T^{4} \)
$5$ \( 1 + 8 T^{2} + 25 T^{4} \)
$7$ \( ( 1 + 2 T + 2 T^{2} + 14 T^{3} + 49 T^{4} )^{2} \)
$11$ \( ( 1 - 20 T^{2} + 121 T^{4} )^{2} \)
$13$ \( ( 1 + 169 T^{4} )^{2} \)
$17$ \( ( 1 - 16 T^{2} + 289 T^{4} )( 1 + 16 T^{2} + 289 T^{4} ) \)
$19$ \( ( 1 - 22 T^{2} + 361 T^{4} )^{2} \)
$23$ \( 1 - 158 T^{4} + 279841 T^{8} \)
$29$ \( ( 1 + 8 T^{2} + 841 T^{4} )^{2} \)
$31$ \( ( 1 + 2 T + 31 T^{2} )^{4} \)
$37$ \( ( 1 - 12 T + 72 T^{2} - 444 T^{3} + 1369 T^{4} )^{2} \)
$41$ \( ( 1 - 50 T^{2} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 - 12 T + 72 T^{2} - 516 T^{3} + 1849 T^{4} )^{2} \)
$47$ \( ( 1 + 2209 T^{4} )^{2} \)
$53$ \( ( 1 - 56 T^{2} + 2809 T^{4} )( 1 + 56 T^{2} + 2809 T^{4} ) \)
$59$ \( ( 1 + 20 T^{2} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 + 6 T + 61 T^{2} )^{4} \)
$67$ \( ( 1 + 8 T + 32 T^{2} + 536 T^{3} + 4489 T^{4} )^{2} \)
$71$ \( ( 1 + 58 T^{2} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 - 6 T + 73 T^{2} )^{2}( 1 + 16 T + 73 T^{2} )^{2} \)
$79$ \( ( 1 - 122 T^{2} + 6241 T^{4} )^{2} \)
$83$ \( 1 - 13294 T^{4} + 47458321 T^{8} \)
$89$ \( ( 1 + 170 T^{2} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 - 6 T + 18 T^{2} - 582 T^{3} + 9409 T^{4} )^{2} \)
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